2ME203.pptx

School of Applied Engineering, Innovation and
Technology
Presentation By: MR. ILEKA
Topic :Dimensioning
MAY- 2023
Unit: 2ME/EEE103/2CE107 – Engineering Drawing

Chapter 4: Control Volume Analysis Using
Energy
4.1 Conservation of Mass for a Control Volume
4.2 Conservation of Energy for a Control Volume
4.3 Analyzing Control Volumes at Steady State
4.4 Transient Analysis

Chapter 4: Control Volume Analysis Using
Energy
• The objective of this chapter is to develop and illustrate the
use of the control volume forms of the conservation of
mass and conservation of energy principles.
• As in the case of a closed system, energy transfer across the
boundary of a control volume can occur by means of work
and heat.
• In addition, another type of energy transfer is —the energy
accompanying mass as it enters or exits.

• In this section an expression of the conservation of mass
principle for control volumes is developed and illustrated.
• The one-dimensional flow model is introduced.

4.1.1 Developing the Mass Rate
Balance
• A control volume with mass flowing
in at i and flowing out at e, is show
below.
• When the mass rate balance applied
to such a control volume, the
conservation of mass principle states

• Denoting the mass contained within the control volume at
time t by,
• The conservation of mass principle can be expressed in
symbols as
• where
• is the time rate of change of mass within the control
volume,
• 𝒎𝒊 𝒂𝒏𝒅 𝒎𝒆 are the mass flow rates at the inlet and exit,
respectively.

• In case, there are several locations on the boundary through
which mass enters or exits. This can be accounted for by
summing, as follows
• The mass rate balance for control volumes with several
inlets and exits.

D E V E LO P I N G T H E CO N T R O L
VO L U M E MAS S BALANCE
• A system consisting of a fixed quantity
of matter m that occupies different
regions at time t and a later time t + ∆t.
• In a time interval ∆t all the mass in
region i crosses the control volume
boundary, while some of the mass, me,
initially contained within the control
volume exits to fill the region e .
• Although the mass in regions i and e as
well as in the control volume differ from
time t to t + ∆t, the total amount of
mass is constant.

• It states that the change in mass of the control volume
during time interval ∆t equals the amount of mass that
enters minus the amount of mass that exits.
• divide by ∆t to obtain
• Then, in the limit as ∆t goes to zero

EVALUATING THE MASS FLOW RATE
• consider a small quantity of matter
flowing with velocity V across an
incremental area dA in a time interval ∆t,
as below.
• The volume of the matter crossing dA
during the time interval ∆t is an oblique
cylinder with a volume equal to the
product of the area of its base dA and its
altitude Vn ∆t .
• Vn is the component of the relative
velocity normal to dA in the direction of
flow.

• Multiplying by the density gives the amount of mass that
crosses dA in time ∆t
• By dividing both sides of this equation by ∆t and taking the
limit as ∆t goes to zero, the mass flow rate across
incremental area dA is:

4.1.2 Forms of the Mass Rate Balance
• it is convenient to apply the mass balance in forms which
suit to particular objectives.
• Forms:
ONE-DIMENSIONAL FLOW FORM
STEADY-STATE FORM
INTEGRAL FORM

ONE-DIMENSIONAL FLOW FORM
• When a flowing stream of matter entering or exiting a
control volume obeys to the following idealizations, the
flow is said to be one-dimensional:
 The flow is normal to the boundary at locations where
mass enters or exits the control volume.
 All intensive properties, including velocity and density, are
uniform with position (bulk average values) over each inlet
or exit area through which matter flows.

• for one-dimensional flow, the mass flow rate becomes
• or in terms of specific volume
• The product AV is the volumetric flow rate.
• The volumetric flow rate is expressed in units of m3/s.
• the conservation of mass principle for control volumes of
one-dimensional flow at the inlet and exits

STEADY-STATE FORM
• Many engineering systems can be idealized as being at
steady state, which means that all properties are
unchanging in time.
• For a control volume at steady state, the identity of the
matter within the control volume changes continuously, but
the total amount present at any time remains constant. So
• That is, the total incoming and outgoing rates of mass flow
are equal.

INTEGRAL FORM
• The total mass contained within the control volume at an
instant t can be related to the local density as follows:
• where the integration is over the volume at time t.
• The mass rate balance can be written as:
• The product is known as the mass flux, which gives the
time rate of mass flow per unit of area.

• E X A M P L E 4 . 1 Feedwater Heater at Steady State
• A feedwater heater operating at steady state has two inlets
and one exit. At inlet 1, water vapor enters at p1 7 bar, T1
200C with a mass flow rate of 40 kg/s. At inlet 2, liquid
water at p2 7 bar, T2 40C enters through an area A2 25 cm2
. Saturated liquid at 7 bar exits at 3 with a volumetric flow
rate of 0.06 m3 /s. Determine the mass flow rates at inlet 2
and at the exit, in kg/s, and the velocity at inlet 2, in m/s.

4.2 Conservation of Energy for a Control
Volume
4.2.1 Developing the Energy Rate
Balance for a Control Volume
• Consider the control Volume shown
below
• the conservation of energy principle
applied to a control volume states:
• For the one-inlet, one-exit control
volume with one-dimensional flow
the energy rate balance is

Volume
• where
• is the energy of the control volume at time t.
• The underlined terms account for the rates of transfer of
internal, kinetic, and potential energy of the entering and
exiting streams.

Volume
EVALUATING WORK FOR A CONTROL VOLUME
• Because work is always done on or by a control volume
where matter flows across the boundary, it is convenient to
separate the work term into two contributions:
One contribution is the work associated with the fluid
pressure as mass is introduced at inlets and removed at
exits
The other contribution, denoted by includes all other
work effects, such as those associated with rotating shafts,
displacement of the boundary, and electrical effects.

Volume
• Consider the work at an exit e associated with the pressure of
the flowing matter.
• The rate at which work is done at the exit by the normal force
(normal to the exit area in the direction of flow) due to pressure
is the product of the normal force, peAe, and the fluid velocity,
Ve.
• That is
• where pe is the pressure, Ae is the area, and Ve is the velocity at
exit e.
• A similar expression can be written for the rate of energy
transfer by work into the control volume at inlet i.

Volume
• With these considerations, the work term of the energy rate
equation, can be written as
• the term at the inlet has a negative sign because energy is
transferred into the control volume there.
• A positive sign at the work term at the exit because energy
is transferred out of the control volume there.
• With
• where 𝒎𝒊 and 𝒎𝒆 are the mass flow rates and vi and ve are
the specific volumes evaluated at the inlet and exit,
respectively.

Volume
• are referred as flow work.

Volume
4.2.2 Forms of the Control Volume Energy Rate Balance
• Substituting
• In
• and collecting all terms referring to the inlet and the exit,
the control volume energy rate balance becomes:
• With h = u + pv, the energy rate balance becomes

Volume
• In case where there are several locations on the boundary
through which mass enters or exits.

4.3 Analyzing Control Volumes at Steady
State
4.3.1 Steady-State Forms of the Mass and Energy Rate
Balances
• For a control volume at steady state:
the conditions of the mass within the control volume and
at the boundary do not vary with time.
The mass flow rates and the rates of energy transfer by
heat and work are also constant with time.
No accumulation of mass within the control volume, so

State
• the mass rate balance
• Furthermore, at steady state
• the Energy rate balance

State
• For the process between state 1 and 2,
• 𝑖 = 1 𝑎𝑛𝑑 𝑒 = 2
• Then
• dividing by the mass flow rate

State
4.3.3 Engineering systems: components :
• Control Volumes at Steady State of
NOZZLES AND DIFFUSERS
Turbine
COMPRESSORS AND PUMPS
Heat exchangers (condensers, evaporators)
THROTTLING DEVICES

State
a) NOZZLES AND DIFFUSERS
• A nozzle is a flow passage of varying cross-sectional area in
which the velocity of a gas or liquid increases in the
direction of flow.
• In a diffuser, the gas or liquid decelerates in the direction
of flow.

State
• A nozzle and diffuser can be combined, example in a wind-
tunnel test facility.

State
• For nozzles and diffusers, the only work is flow work at
locations where mass enters and exits the control volume,
• At steady state the mass and energy rate balances reduce,
respectively, to
• By combining these into a single expression and dropping
the potential energy change from inlet to exit

State
E X A M P L E 4 . 3 Calculating Exit Area of a Steam Nozzle

b) TURBINES
• A turbine is a device in which work is developed as a result
of a gas or liquid passing through a set of blades attached to
a rotating shaft.
Types
A axial-flow steam
 A gas turbine
A hydraulic turbine

• Turbines are widely used in vapor power plants, gas turbine
power plants, and aircraft engines.
• In these applications, superheated steam or a gas enters
the turbine and expands to a lower exit pressure as work is
developed.

• A hydraulic turbine installed in a dam.
• In this application, water falling through the propeller
causes the shaft to rotate and work is developed.

• For a turbine at steady state the mass and energy rate
balances reduce to:
• When gases are under consideration, the potential energy
change is ignored.
• The kinetic energy change is usually small enough to be
neglected too.
• ∆KE = 0
• ∆PE = 0

• E X A M P L E 4 . 4 Calculating Heat Transfer from a Steam
Turbine

c) COMPRESSORS AND PUMPS
• Compressors are devices in which work is done on a gas passing
through them in order to increase the pressure.
• In pumps, the work input is used to change the state of a liquid
passing through.
Types:
A reciprocating compressor
A rotating compressor
Rotating compressors:
• an axial-flow compressor,
• a centrifugal compressor,
• a Roots type.

• The mass and energy rate balances reduce for compressors
and pumps at steady state, are as for the turbines.
• The changes in specific kinetic and potential energies from
inlet to exit are often small relative to the work done per
unit of mass passing through the device.

E X A M P L E 4 . 5 Calculating Compressor Power

d) HEAT EXCHANGERS
• Devices that transfer energy between fluids at different
temperatures by heat transfer modes are called heat
exchangers.
• The common type of heat exchanger:
 mixed streams, in which hot and cold streams are mixed
directly.
separated streams, in which a gas or liquid is separated from
another gas or liquid by a wall through which energy is
conducted. (recuperators)
Counterflow and parallel tube-within-a-tube configurations
cross-flow, as in automobile radiators, and multiple-pass shell-
and-tube condensers and evaporators.

• The only work interaction at the boundary of a control
volume enclosing a heat exchanger is flow work at the
places where matter enters and exits,
• 𝑊
𝑐𝑣=0
• 𝑄𝑐𝑣=0
• ∆KE = 0
• ∆PE = 0

• E X A M P L E 4 . 7 Power Plant Condenser

THROTTLING DEVICES
• A significant reduction in pressure can be achieved simply
by introducing a restriction into a line through which a gas
or liquid flows.
• This is commonly done by means of a partially opened
valve or a porous plug.
• For a control volume enclosing such a device, the mass and
energy rate balances reduce at steady state to

• 𝑄𝑐𝑣=0
• ∆PE = 0
• the mass and energy rate balances combine to give
• in most cases that ∆KE = 0 for gas or liquid between inlet
and exit.
• When the flow through a valve or other restriction is
idealized in this way, the process is called a throttling
process.

• E X A M P L E 4 . 9 Measuring Steam Quality

SYSTEM INTEGRATION
• Turbine
• COMPRESSORS AND PUMPS
• Heat exchangers (condensers, evaporators)
• THROTTLING DEVICES
• All the types of components that we have looked at, can be
combined to form an integrated system (called system
integration)
• One of the system integration: the simple power plant.
• This system consists of four components in series, a turbine,
condenser, pump, and boiler.

4.3 Analyzing Control Volumes at
Steady State
• Simple vapor power plant.

E X A M P L E 4 . 1 0 Waste Heat Recovery System

• Many devices undergo periods of transient operation in
which the state changes with time.
Examples include:
 the startup or shutdown of turbines, compressors, and
motors.
vessels being filled or emptied,
• Because property values, work and heat transfer rates, and
mass flow rates may vary with time during transient
operation, the steady-state assumption is not appropriate
when analyzing such cases.

MASS BALANCE
• the control volume mass balance is placed in a form that is
suitable for transient analysis.
• It is done by integrating the mass rate balance, from time 0
to a final time t.
• That is
• This takes the form

• Introducing the following symbols for the underlined terms
• the mass balance becomes
• the change in the amount of mass contained in the control
volume equals the difference between the total incoming and
outgoing amounts of mass.

ENERGY BALANCE
• integrate the energy rate balance,
• ∆KE = 0
• ∆PE = 0
• underlined term account for the energy carried in at the
inlets and out at the exits.

• For the special case where the states at the inlets and exits
are constant with time, the respective
• Then the energy equ. becomes:
• specific enthalpies, hi and he, are taken to be constant.

• Another special case is when the intensive properties
within the control volume are uniform with position at each
instant.
• Accordingly, the specific volume and the specific internal
energy are uniform throughout and can depend only on
time, that is v(t) and u(t).
• Thus

• E X A M P L E 4 . 1 1 Withdrawing Steam from a Tank at
Constant Pressure

2ME203.pptx

Recomendados

Recomendados

Mais conteúdo relacionado

Semelhante a 2ME203.pptx

Semelhante a 2ME203.pptx (20)

Último

Último (20)

2ME203.pptx